Interval-Valued Neutrosophic Graph Structures

Punjab University Journal of Mathematics (ISSN 1016-2526) Vol. 50(1)(2018) pp. 113-137

Interval-Valued Neutrosophic Graph Structures Muhammad Akram Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan.

Email: m.akram@pucit.edu.pk Muzzamal Sitara Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan.

Email: muzzamalsitara@gmail.com Received: 16 May, 2017 / Accepted: 11 October, 2017 / Published online: 16 November, 2017 Abstract. In this research article, we introduce certain notions of interval-valued neutrosophic graph structures. We elaborate the concepts of interval-valued neutrosophic graph structures with examples. Moreover, we discuss the concept of ϕ-complement of an interval-valued neutrosophic graph structure. Finally, we present some related properties, including ϕ-complement, totallyself complementary and totally-strong-self complementary, of intervalvalued neutrosophic graph structures.

AMS (MOS) Subject Classification Codes: 35S29, 40S70, 25U09 Key Words: Graph structure, Interval-valued neutrosophic graph structure, ϕcomplement.

1. I NTRODUCTION Zadeh [33] introduced interval-valued fuzzy set theory which is an extension of fuzzy set theory [32]. Membership degrees in an interval-valued fuzzy set are intervals rather than numbers and uncertainty is reflected by length of interval membership degree. Zhan et al. [35, 36] applied the concept of interval-valued fuzzy sets to algebraic structures. For representing vagueness and uncertainty Atanassov [10] proposed an extension of fuzzy sets by adding a new component, called intuitionistic fuzzy sets. The concept of intuitionistic fuzzy sets is more meaningful and inventive due to the presence of degree of truth, indeterminacy

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and falsity membership. The intuitionistic fuzzy sets have more describing possibilities as compared to fuzzy sets. The hesitation margin of an intuitionistic fuzzy set is its uncertainty by default and sum of truth-membership degree and falsitymembership degree does not exceeds unity. In many phenomenons like information fusion, uncertainty and indeterminacy is doubtlessly quantified. Smarandache [24, 25] proposed the idea of neutrosophic sets, he mingled tricomponent logic, non-standard analysis, and philosophy. “It is a branch of philosophy which studies the origin, nature and scope of neutralities as well as their interactions with different ideational spectra”. For convenient and advantageous usage of neutrosophic sets in science and engineering, Wang et al. [29] proposed the notion of singlevalued neutrosophic(SVNS)sets, whose three independent components have values in standard unit interval [0, 1]. Neutrosophic set theory being a generalization of fuzzy set theory and intuitionistic fuzzy set theory, is more practical, advantageous and applicable in various fields, including medical diagnosis, control theory, topology, decision-making problems and in many more real-life problems. Wang et al. [30] proposed the notion of interval-valued neutrosophic sets, which is more precise and flexible than the single-valued neutrosophic sets. An interval-valued neutrosophic set is a generalization of the notion of single-valued neutrosophic set, in which three independent components (t, i, f ) are intervals which are subsets of standard unit interval [0, 1].

On the basis of Zadeh’s fuzzy relations [35] Kaufmann proposed fuzzy graph [18]. Rosenfeld [22] discussed fuzzy analogue of various graph-theoretic ideas.

Later on, Bhattacharya [11] gave some remarks on fuzzy graphs, and some operations on fuzzy graphs were introduced by Mordeson and Peng [19]. Mordeson and Nair presented a valuable contribution on fuzzy graphs as well as fuzzy hypergraphs in [20]. Mathew and Sunitha [26, 27] discussed arcs and strong cycles in fuzzy graphs. On the other hand, Dinesh and Ramakrishnan [14] defined fuzzy graph structures and discussed its properties. Akram and Akmal [7] proposed the notion of bipolar fuzzy graph structures. Akram et al. [1, 2, 3, 4] have introduced several concepts on interval-valued fuzzy graphs and interval-valued neutrosophic graphs. Akram and Shahzadi [8] introduced the notion of neutrosophic soft graphs with applications. Recently, Akram and Nasir [5, 6] considered interval-valued neutrosophic graphs. In this research article, we introduce certain notions of interval-valued neutrosophic graph structures. We elaborate the concepts of interval-valued neutrosophic graph structures with examples. Moreover, we discuss the concept of ϕ-complement of an interval-valued neutrosophic graph structure.

2. I NTERVAL -VALUED N EUTROSOPHIC G RAPH S TRUCTURES Sampathkumar [23] introduced the graph structure which is a generalization of undirected graph and is quite useful in studying some structures, including, graphs, signed graphs, labelled graphs and edge colored graphs.

Definition 2.1. [23] A graph structure G = (U, R1 , . . . , Rt ) consists of a nonempty set U together with relations R1 , R2 , . . . , Rt on U which are mutually disjoint such that each Rj , 1 ≤ j ≤ t, is symmetric and irreflexive.

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One can represent a graph structure G = (U, R1 , . . . , Rt ) in the plane just like a graph where each edge is labelled as Rj , 1 ≤ j ≤ t.

Example 2.2. Let U = {r1 , r2 , r3 , r4 , r5 }. Let R1 = {(r1 , r2 ), (r3 , r4 ), (r1 , r4 )}, R2 = {(r1 , r3 ), (r1 , r5 )}, R3 = {(r2 , r3 ), (r4 , r5 )} be mutually disjoint, symmetric and irreflexive relations on set U . Thus G = (U, R1 , R2 , R3 ) is a graph structure and is represented in plane as a graph where each edge is labelled as R1 , R2 or R3 . r1 b

R1

r2

R2

b

b

R1

R2

r5

R3

R3 b b

r3

R1

r4

F IGURE 1. Graph structure G = (U, R1 , R2 , R3 )

Definition 2.3. [30, 31] The interval-valued neutrosophic set I on set U is defined by I = {(r, [t− (r), t+ (r)], [i− (r), i+ (r)], [f − (r), f + (r)]) : r ∈ U }, where, t− , t+ , i− , i+ , f − , and f + are functions from U to [0, 1] such that: t− (r) ≤ t+ (r), i− (r) ≤ i+ (r) and f − (r) ≤ f + (r) for all r ∈ U . ˇ iv = (I, I1 , I2 , . . . , It ) is called an interval-valued neutrosophic Definition 2.4. G graph structure(IVNGS) of graph structure G = (U, R1 , R2 , . . . , Rt ) if I = {(r, [t− (r), t+ (r)], [i− (r), i+ (r)], [f − (r), f + (r)]) : r ∈ U } + − + − + and Ij = { ((r, s), [t− j (r, s), tj (r, s)], [ij (r, s), ij (r, s)], [fj (r, s), fj (r, s)]) : (r, s) ∈ Rj } are interval-valued neutrosophic(IVN) sets on U and Rj , respectively, such that: + + − − t+ (1) t− j (r, s) ≤ min{t (r), t (s)}, j (r, s) ≤ min{t (r), t (s)}, − − + + (2) i− i+ j (r, s) ≤ min{i (r), i (s)}, j (r, s) ≤ min{i (r), i (s)}, − + − − + (3) fj (r, s) ≤ min{f (r), f (s)}, fj (r, s) ≤ min{f (r), f + (s)}, + − + − + where, t− j , tj , ij , ij , fj , and fj are functions from Rj to [0, 1] such that − + − − + tj (r, s) ≤ tj (r, s), ij (r, s) ≤ i+ j (r, s) and fj (r, s) ≤ fj (r, s) for all (r, s) ∈ Rj .

In this paper we will use rs in place of ordered pair (r, s) which represents an edge between vertices r and s.

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F IGURE 2. Interval-valued neutrosophic graph structure

Example 2.5. Consider the graph structure G = (U, R1 , R2 ) such that U = {r1 , r2 , r3 , r4 }, R1 = {r1 r3 , r1 r2 , r3 r4 }, R2 = {r1 r4 , r2 r3 }. By defining intervalvalued neutrosophic sets I, I1 and I2 on U , R1 and R2 , respectively, we draw an IVNGS as shown in Fig. 2. ˇ iv = (I, I1 , I2 , . . . , It ) be an IVNGS of graph structure(GS) Definition 2.6. Let G ˇ iv = (I ′ , I1′ , I2′ , . . . , It′ ) is an IVNGS of G such that G = (U, R1 , R2 , . . . , Rt ). If H t′− (r) ≤ t− (r), i′− (r) ≤ i− (r), f ′− (r) ≤ f − (r), t′+ (r) ≤ t+ (r), i′+ (r) ≤ i+ (r), f ′+ (r) ≤ f + (r) , − ′− − ′− − t′− j (rs) ≤ tj (rs), ij (rs) ≤ ij (rs), fj (rs) ≤ fj (rs), ′+ + ′+ + ′+ + tj (rs) ≤ tj (rs), ij (rs) ≤ ij (rs), fj (rs) ≤ fj (rs), for all r ∈ U and rs ∈ Rj , j = 1, 2, . . . , t.

ˇ iv is called an interval-valued neutrosophic(IVN) subgraph-structure of Then H ˇ iv .

IVNGS G ˇ iv = (I ′ , I ′ , I ′ ) of graph structure G = Example 2.7. Consider an IVNGS H 1 2 (U, R1 , R2 ) as illustrated in Fig. 3. Through direct calculations, it is shown that ˇ iv is an IVN subgraph-structure of IVNGS G ˇ iv shown in Fig. 2.

H

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F IGURE 4. IVN induced subgraph-structure

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F IGURE 3. Interval-valued neutrosophic subgraph-structure

ˇ iv = (I ′ , I ′ , I ′ , . . . , I ′ ) is called an IVN induced Definition 2.8. An IVNGS H 1 2 t ˇ iv by Q ⊆ U if subgraph-structure of IVNGS G t′− (r) = t− (r), i′− (r) = i− (r), f ′− (r) = f − (r), t′+ (r) = t+ (r), i′+ (r) = i+ (r), f ′+ (r) = f + (r), − ′− − ′− − ′+ + t′− j (rs) = tj (rs), ij (rs) = ij (rs), fj (rs) = fj (rs), tj (rs) = tj (rs), ′+ + ′+ + ij (rs) = ij (rs), fj (rs) = fj (rs), for all r, s ∈ Q, j = 1, 2, . . . , t. ˇ iv = (I ′ , I1′ , I2′ ) of graph structure G = (U, R1 , R2 ) Example 2.9. An IVNGS H ˇ iv = (I, I1 , I2 ) shown in Fig. 4 is an IVN induced subgraph-structure of IVNGS G represented in Fig. 2. ˇ iv = (I ′ , I ′ , I ′ , . . . , I ′ ) is called IVN spanning Definition 2.10. An IVNGS H 1 2 t ˇ subgraph-structure of IVNGS Giv = (I, I1 , I2 , . . . , It ) if I ′ = I and − ′− − ′− − t′− j (rs) ≤ tj (rs), ij (rs) ≤ ij (rs), fj (rs) ≤ fj (rs),

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M. Akram and M. Sitara

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F IGURE 5. IVN spanning subgraph-structure + ′+ + ′+ + t′+ j (rs) ≤ tj (rs), ij (rs) ≤ ij (rs), fj (rs) ≤ fj (rs), j = 1, 2, . . . , t.

ˇ iv = (I ′ , I ′ , I ′ ) shown in Fig. 5 is an IVN spanning Example 2.11. An IVNGS H 1 2 ˇ iv = (I, I1 , I2 ) represented in Fig. 2. subgraph-structure of IVNGS G ˇ iv = (I, I1 , I2 , . . . , It ) be an IVNGS. Then edge rs ∈ Ij is Definition 2.12. Let G called an IVN Ij -edge or in short an Ij -edge if − − + + t− j (rs) > 0 or ij (rs) > 0 or fj (rs) > 0 or tj (rs) > 0 or ij (rs) > 0 or + fj (rs) > 0 or all of conditions are satisfied. Hence support of Ij is defined as; supp(Ij ) = − − {rs ∈ Ij : t− j (rs) > 0} ∪ {rs ∈ Ij : ij (rs) > 0} ∪ {rs ∈ Ij : fj (rs) > 0} ∪ + + {rs ∈ Ij : tj (rs) > 0} ∪ {rs ∈ Ij : ij (rs) > 0} ∪ {rs ∈ Ij : fj+ (rs) > 0}, j = 1, 2, . . . , t.

ˇ iv = (I, I1 , I2 , . . . , It ) is a sequence Definition 2.13. An Ij -path in an IVNGS G r1 , r2 , . . . , rt of distinct vertices (except rt = r1 ) in U such that rj−1 rj is an IVN Ij -edge for all j = 2, 3, . . . , t. ˇ iv = (I, I1 , I2 , . . . , It ) is Ij -strong for any j ∈ Definition 2.14. An IVNGS G {1, 2, . . . , t} if − − − − − t− j (rs) = min{t (r), t (s)}, ij (rs) = min{i (r), i (s)}, + + fj− (rs) = min{f − (r), f − (s)}, t+ j (rs) = min{t (r), t (s)}, + + + + + + ij (rs) = min{i (r), i (s)}, fj (rs) = min{f (r), f (s)}, ˇ iv is Ij -strong for all j ∈ {1, 2, . . . , t}, then G ˇ iv is for all rs ∈ supp(Ij ). If G called a strong IVNGS.

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r6 ([0.1, 0.2], [0.5, 0.6], [0.4, 0.5])

F IGURE 6. Strong IVNGS ˇ iv = (I, I1 , I2 , I3 ) as shown in Fig. 6. G ˇ iv Example 2.15. Consider an IVNGS G is a strong IVNGS, since it is I1 , I2 and I3 strong. ˇ iv = (I, I1 , I2 , . . . , It ) is called complete IVNGS, Definition 2.16. An IVNGS G if ˇ iv is a strong IVNGS. (1) G (2) Supp(Ij ) 6= ∅, for all j = 1, 2, . . . , t.

(3) For all r, s ∈ U , rs is an Ij − edge for some j. ˇ iv = (I, I1 , I2 , I3 ) be an IVNGS of graph structure G = Example 2.17. Let G (U, R1 , R2 , R3 ) and it is shown in Fig.7. Where, U = {r1 , r2 , r3 , r4 , r5 , r6 }, R1 = {r1 r6 , r1 r2 , r2 r4 , r2 r5 , r2 r6 , r4 r5 }, R2 = {r4 r3 , r5 r6 , r1 r4 }, R3 = {r1 r5 , r5 r3 , r2 r3 , r1 r3 , r4 r6 }. By direct ˇ iv is a strong IVNGS. Moreover, supp(I1 ) 6= ∅, calculations, we can show that G supp(I2 ) 6= ∅, supp(I3 ) 6= ∅ and each pair rj rk of nodes in U , is either an ˇ iv is a complete IVNGS, that is, I1 − edge or I2 −edge or I3 − edge. Hence G I1 I2 I3 −complete IVNGS. ˇ iv = (I, I1 , I2 , . . . , It ) be an IVNGS. The truth strength Definition 2.18. Let G + − [t .PIj , t .PIj ], indeterminacy strength [i− .PIj , i+ .PIj ] and falsity strength [f − .PIj , f + .PIj ] of an Ij -path, PIj = r1 , r2 , . . . , rn are defined as: n n V V [t+ [t− .PIj , t+ .PIj ] = [ [t− Ij (rk−1 rk )]], Ij (rk−1 rk )], [i− .PIj , i+ .PIj ] = [

k=2 n V

[i− Ij (rk−1 rk )],

k=2

k=2 n V

k=2

[i+ Ij (rk−1 rk )]],

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ˇ iv = (I, I1 , I2 ) F IGURE 8. IVNGS G [f − .PIj , f + .PIj ] = [

n V

k=2

[fI−j (rk−1 rk )],

n V

k=2

[fI+j (rk−1 rk )]].

ˇ iv = (I, I1 , I2 ) of Graph structure G = Example 2.19. Consider an IVNGS G (U, R1 , R2 ) as shown in Fig. 8. For I2 -path PI2 = r1 , r5 , r3 , r6 , [t− .PI2 , t+ .PI2 ] = [0.2, 0.3], [i− .PI2 , i+ .PI2 ] = [0.1, 0.2] and [f − .PI2 , f + .PI2 ] = [0.3, 0.4]. ˇ iv = (I, I1 , I2 , . . . , It ) be an IVNGS. Then Definition 2.20. Let G

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• Ij - Truth strength of connectedness between two nodes r and s is defined by: W +i W −i +∞ {tIj (rs)}] such that {tIj (rs)}, [t−∞ Ij (rs), tIj (rs)] = [ i≥1 −(i−1)

i≥1

+(i−1)

−(1)

+(1)

+i ◦tIj )(rs)] for i ≥ 2 ◦tIj )(rs), (tIj [t−i Ij (rs), tIj (rs)] = [(tIj −2 +2 −1 −1 +1 and [tIj (rs), tIj (rs)] = [(tIj ◦ tIj )(rs), (tIj ◦ t+1 Ij )(rs)] W W +1 −1 +1 = [ (t−1 (ry) ∧ t (ys)), (t (ry) ∧ t (ys))].

Ij Ij Ij Ij y

y

• Ij - Indeterminacy strength of connectedness between two nodes r and s is defined by: W +i W −i +∞ {iIj (rs)}] such that {iIj (rs)}, [i−∞ Ij (rs), iIj (rs)] = [ i≥1 i≥1 +(i−1) +(1) −(i−1) −(1) +i −i ◦iIj )(rs)] ◦iIj )(rs), (iIj [iIj (rs), iIj (rs)] = [(iIj +1 +1 −1 −1 +2 ◦ i )(rs), (i ◦ i (rs)] = [(i (rs), i and [i−2 Ij )(rs)] Ij Ij Ij Ij j W I−1 W −1 +1 +1 = [ (iIj (ry) ∧ iIj (ys)), (iIj (ry) ∧ iIj (ys))]. y y

for i ≥ 2

• Ij - Falsity strength of connectedness between two nodes r and s is defined by: W +i W {fIj (rs)}] such that (rs)}, [fI−∞ (rs), fI+∞ (rs)] = [ {fI−i j j j i≥1 −(i−1)

i≥1

−(1)

+(i−1)

+(1)

[fI−i ◦ fIj )(rs)] for (rs), fI+i ◦ fIj )(rs), (fIj (rs)] = [(fIj j j +1 −1 −1 +2 −2 )(rs)] i ≥ 2 and [fIj (rs), fIj (rs)] = [(fIj ◦ fIj )(rs), (fIj ◦ fI+1 j W −1 W +1 −1 +1 = [ (fIj (ry) ∧ fIj (ys)), (fIj (ry) ∧ fIj (ys))]. y

y

ˇ iv = (I, I1 , I2 , . . . , It ) is called an Ij -cycle if (supp(I), Definition 2.21. An IVNGS G supp(I1 ),supp(I2 ), . . . , supp(It )) is an Ij − cycle.

ˇ iv = (I, I1 , I2 , . . . , It ) is an IVN fuzzy Ij -cycle Definition 2.22. An IVNGS G ˇ ˇ iv such that: (for some j) if Giv is an Ij -cycle and no unique Ij -edge rs exists in G − + − [tIj (rs), tIj (rs)] = [min{tIj (uv) : uv ∈ Ij = supp(Ij )}, min{t+ Ij (uv) : uv ∈ Ij = supp(Ij )}] or − − [iIj (rs), i+ Ij (rs)] = [min{iIj (uv) : uv ∈ Ij = supp(Ij )}, + min{iIj (uv) : uv ∈ Ij = supp(Ij )}] or [fI−j (rs), fI+j (rs)] = [min{fI−j (uv) : uv ∈ Ij = supp(Ij )}, min{fI+j (uv) : uv ∈ Ij = supp(Ij )}].

ˇ iv = (I, I1 , I2 ) of graph structure G = Example 2.23. Consider an IVNGS G (U, R1 , R2 ) as shown in Fig. 9.

122

M. Akram and M. Sitara

0 .4

,0

.5 ]

)

]) 0.4 .3, 0 [ , ] 0.4 ]) .3, 0.4 .3, ], [0 0 3 [ . , 2, 0 , 0.4] [0.

[0.3 I 1( 3], . ,0 2 . ([0

b

[0.

1, 0.2 ], [ 0.3 ,0 .4] , [0 .3, 0.4 ])

], [

b

r1 ([0.3, 0.4], [0.4, 0.5], [0.5, 0.6])

I2 (

,0

I2 ([0.2, 0.3], [0.2, 0.3], [0.3, 0.4])

[0.1 I 1( b

r3 ([0.10.2], [0.4, 0.5], [0.5, 0.6])

b

r4 ([0.2, 0.3], [0.2, 0.3], [0.4, 0.5]) 5]) I2 ([0 , 0. .2 , [0.4 0 .3 , ] 3 ], [ , 0.

0 .2 2 .

[0 ,0 , ] .3 .2

r2

I2 ([0.1, 0.2], [0.4, 0.5], [0.5, 0.6])

F IGURE 9. Interval-valued neutrosophic fuzzy I2 -cycle

ˇ iv is an I2 -cycle, that is, r1 − r4 − r2 − r3 − r1 and no unique This IVNGS G ˇ iv satisfying following condition: I2 -edge rs exists in G − + [tI2 (rs), tI2 (rs)] = [min{t− I2 (uv) : uv ∈ I2 = supp(I2 )}, (uv) : uv ∈ I = supp(I min{t+ 2 2 )}] or I2 + − [i− (rs), i (rs)] = [min{i (uv) : uv ∈ I2 = supp(I2 )}, I2 I2 I2 + min{iI2 (uv) : uv ∈ I2 = supp(I2 )}] or [fI−2 (rs), fI+2 (rs)] = [min{fI−2 (uv) : uv ∈ I2 = supp(I2 )}, min{fI+2 (uv) : uv ∈ I2 = supp(I2 )}]. ˇ iv = (I, I1 , I2 , . . . , It ) be an IVNGS and ‘r’ be a vertex of Definition 2.24. Let G ˇ iv . If (I ′ , I ′ , I ′ , . . . , It′ ) is an IVN subgraph-structure of G ˇ iv induced by U \{r} G 1 2 such that for all u 6= r, v 6= r − − − − − t− I ′ (r) = iI ′ (r) = fI ′ (r) = tI ′ (ru) = iI ′ (ru) = fI ′ (ru) = 0, j

j

j

j

j

j

+ + + + + t+ I ′ (r) = iI ′ (r) = fI ′ (r) = tI ′ (ru) = iI ′ (ru) = fI ′ (ru) = 0, + − + + − + [t− [i− I ′ (u), tI ′ (u)] = [tI (u), tI (u)], I ′ (u), iI ′ (u)] = [iI (u), iI (u)], − + − + [fI ′ (u), fI ′ (u)] = [fI (u), fI (u)], + − + + + − [t− (uv), t (uv)] = [t− ′ ′ Ij (uv), tIj (uv)], [iIj′ (uv), iIj′ (uv)] = [iIj (uv), iIj (uv)], Ij Ij [fI−′ (uv), fI+′ (uv)] = [fI−j (uv), fI+j (uv)]. j j

ˇ iv , then vertex r is an IVN fuzzy Ij cut-vertex, if for all edges ru, uv ∈ G −∞ −∞ +∞ +∞ −∞ (1) tIj (uv) > tI ′ (uv), t+∞ Ij (uv) > tI ′ (uv), [tIj (uv), tIj (uv)] ∩ j

+∞ [t−∞ I ′ (uv), tI ′ (uv)] = ∅ j

j

j

−∞ +∞ +∞ +∞ −∞ (2) i−∞ Ij (uv) > iI ′ (uv), iIj (uv) > iI ′ (uv), [iIj (uv), iIj (uv)] ∩ j

+∞ [i−∞ I ′ (uv), iI ′ (uv)] = ∅ j

j

j

(uv) > fI−∞ (3) fI−∞ (uv) > fI+∞ (uv)] ∩ (uv), fI+∞ (uv), fI+∞ (uv), [fI−∞ ′ ′ j j j j j

[fI−∞ (uv), fI+∞ (uv)] = ∅ ′ ′ j

j

j

for some u, v ∈ U \ {r}. Note that vertex r is an

Interval-Valued Neutrosophic Graph Structures

123

−∞ +∞ +∞ • IVN fuzzy Ij −t cut-vertex, if t−∞ Ij (uv) > tI ′ (uv), tIj (uv) > tI ′ (uv), j

j

+∞ −∞ +∞ [t−∞ Ij (uv), tIj (uv)]∩ [tI ′ (uv), tI ′ (uv)] = ∅ j

j

−∞ +∞ +∞ • IVN fuzzy Ij −i cut-vertex, if i−∞ Ij (uv) > iI ′ (uv), iIj (uv) > iI ′ (uv), j

j

+∞ −∞ +∞ [i−∞ Ij (uv), iIj (uv)]∩ [iI ′ (uv), iI ′ (uv)] = ∅ j

j

(uv) > fI−∞ • IVN fuzzy Ij −f cut-vertex, if fI−∞ (uv) > fI+∞ (uv), fI+∞ (uv), ′ ′ j j j

j

[fI−∞ (uv), fI+∞ (uv)]∩ [fI−∞ (uv), fI+∞ (uv)] = ∅ ′ ′ j j j

j

I2 ([0.1, 0.2], [0.4, 0.5], [0.2, 0.3]) I1 ([0 r6 5, 0 r4 ([0.5, 0.6], [0.7, 0.9], .3, ([0 [0.5, 0.6]) .6] 0.4 .3, , [0 ], [ .7, 0.4 0.2 0.8 ] , ,0 ], [ [0. .3] 0.5 4 ,0 , [0 , 0. .5] 6]) .3, , [0 0.4 I2 ([0.2, 0.3], [0.3, 0.4], [0.4, 0.5]) .3, ]) 0.4 ]) r3 ([0.5, 0.6], [0.7, 0.8], [0.5, 0.6])

r2 ([0.4, 0.5], [0.7, 0.8], [0.5, 0.6]) ([0 b .3, I2 ([0.3, 0.4], [0.6, 0.7], [0.4, 0.5]) 0.4 ], [ 0.5 ,0 .6] , [0 .4, 0, 5]) b b

b

I2 ([0.1, 0.2], [0.4, 0.5], [0.2, 0.3])

], .4 ,0 .3 ([0 I1

b

], .4 ,0 .3 [0

r5 ([0.4, 0, 5], [0.5, 0.6], [0.6, 0.7])

I1

r1 ([0.3, 0.4], [0.6, 0.7], [0.4, 0.5])

I1 ([0.4, 0.5], [0.5, 0.6], [0.4, 0.5])

ˇ iv = (I, I1 , I2 ) of graph structure G = Example 2.25. Consider an IVNGS G (U, R1 , R2 ) as represented in Fig. 10.

]) .5 ,0 .4 [0

I1 ( [0.

b

ˇ iv = (I, I1 , I2 ) F IGURE 10. IVNGS G

ˇ iv = (I ′ , I ′ , I ′ ) is an IVN subgraph-structure of IVNGS G ˇ iv , which is obtained H 1 2 by deleting vertex r2 and shown in Fig. 11.

M. Akram and M. Sitara

r1 ([0 .3, I′ 1 ([ 0.4 0.3 ], [ ,0 0.8 0.6 .

4 ], [ ,0 0.5 r3 ([0.5, 0.6], [0.7, 0.8], [0.5, 0.6]) ], [0 .7] ,0 .

2 , [0 , .6] ) I2′ ([0.2, 0.3], [0.3, 0.4], [0.4, 0.5]) 0.3] . , [0 4, 0.

5] .3, 0.4 ) ])

124

b

r5 ([0.4, 0, 5], [0.5, 0.6], [0.6, 0.7]) b

b

5, 0.6 ]

I2′ ([0.1, 0.2], [0.4, 0.5], [0.2, 0.3])

I ′( 1 [0 .

]) 0.6 ]) .5, .5 , [0 ,0 .9] .4 ,0 [0 ], 0.7 .4 ], [ ,0 0.6 .3 .5, [0 ([0 r4

r6 ([0.3, 0.4], [0.4, 0.5], [0.3, 0.4])

, 4] 0.

3, 0. ′ ([ I1

b

b

, [0 .7,

I2′ ([0.1, 0.2], [0.4, 0.5], [0.2, 0.3])

ˇ iv = (I ′ , I ′ , I ′ ) F IGURE 11. IVNGS H 1 2

The vertex r2 is an IVN fuzzy I1 − i cut-vertex. Since +∞ −∞ +∞ i−∞ I ′ (r4 r5 ) = 0.3, iI1 (r4 r5 ) = 0.5, iI ′ (r4 r5 ) = 0.4, iI1 (r4 r5 ) = 0.6.

1

1

−∞ +∞ +∞ Clearly i−∞ I1 (r4 r5 ) = 0.5 > 0.3 = iI ′ (r4 r5 ), iI1 (r4 r5 ) = 0.6 > 0.4 = iI ′ (r4 r5 ), 1

1

+∞ −∞ +∞ [i−∞ I1 (r4 r5 ), iI1 (r4 r5 )] ∩ [iI ′ (r4 r5 ), iI ′ (r4 r5 )] = [0.5, 0.6] ∩ [0.3, 0.4] = ∅.

1

1

ˇ iv = (I, I1 , I2 , . . . , It ) be an IVNGS and rs be an Ij −edge.

Definition 2.26. Let G ′ ′ ′ ′ ˇ iv , such If (I , I1 , I2 , . . . , It ) is an IVN fuzzy spanning subgraph-structure of G that − − + + + t− I ′ (rs) = iI ′ (rs) = fI ′ (rs) = 0, tI ′ (rs) = iI ′ (rs) = fI ′ (rs) = 0, j

j

j

j

j

j

+ − + − + − [t− I ′ (wx), tI ′ (wx)] = [tIj (wx), tIj (wx)], [iI ′ (wx), iI ′ (wx)] = [iIj (wx), j

j

j

j

i+ Ij (wx)], − [fI ′ (wx), fI+′ (wx)] = [fI−j (wx), fI+j (wx)], j

j

for all edges wx 6= rs, then edge rs is an IVN fuzzy Ij -bridge if −∞ +∞ +∞ +∞ −∞ (1) t−∞ Ij (uv) > tI ′ (uv), tIj (uv) > tI ′ (uv), [tIj (uv), tIj (uv)]∩ j

+∞ [t−∞ I ′ (uv), tI ′ (uv)] = ∅ j

j

j

−∞ +∞ +∞ +∞ −∞ (2) i−∞ Ij (uv) > iI ′ (uv), iIj (uv) > iI ′ (uv), [iIj (uv), iIj (uv)]∩ j

+∞ [i−∞ I ′ (uv), iI ′ (uv)] = ∅ j

j

j

(3) fI−∞ (uv) > fI−∞ (uv), fI+∞ (uv) > fI+∞ (uv), [fI−∞ (uv), fI+∞ (uv)]∩ ′ ′ j j j j j

[fI−∞ (uv), fI+∞ (uv)] = ∅ ′ ′ j

j

for some u, v ∈ U . Note that edge rs is an

j

[0.5, 0.6],

r2 ([0.8, 0.9], [0.7, 0.8], [0.5, 0.7]) b

3, 0.4 ], [ 2 0.6 , , [0 0.3 ,0 .7, I2 ([0.2, 0.3], [0.4, 0.5], [0.3, 0.4]) ], .7] [0.

0.8 , [0 4, ], [ .4, 0 0.5 r3 ([0.5, 0.6], [0.7, 0.8], [0.5, 0.7]) .6 0.5 ], [ ,0 0.3 ]) .6] ) ,0 .4] )

125

[0.5, 0.7])

Interval-Valued Neutrosophic Graph Structures

b

([0 .

r1

I1

([0 .

b

b

5, 0.6 ]

r6 ([0.3, 0.4], [0.4, 0.6], [0.4, 0.5]) ]) .7 ,0 .5 [0 ]) ], .4 .9 ,0 ,0 .3 .7 [0 [0 ], ], .4 .6 ,0 ,0 .5 .3 [0 ([0 r4

], .3 ,0 .2 ([0 I1

I2 ([0.1, 0.2], [0.4, 0.5], [0.2, 0.3])

I1

([0 .

I1 ([0 .7, 0.8 ],

I2 ([0.3, 0.4], [0.6, 0.7], [0.4, 0.5]) I1 ([0 .2, 0.3 ], [ 0.5 ,0 . r5 ([0.7, 0.8], [0.5, 0.6], [0.6, 0.7]) 6], [ 0.3 b b , I2 ([0.1, 0.2], [0.4, 0.5], [0.2, 0.3]) 0.4] )

ˇ iv = (I, I1 , I2 ) F IGURE 12. IVNGS G −∞ +∞ +∞ • IVN fuzzy Ij − t bridge, if t−∞ Ij (uv) > tI ′ (uv), tIj (uv) > tI ′ (uv), j

−∞ +∞ +∞ [t−∞ Ij (uv), tIj (uv)]∩ [tI ′ (uv), tI ′ (uv)] = ∅ j

j

j

−∞ +∞ +∞ • IVN fuzzy Ij − i bridge, if i−∞ Ij (uv) > iI ′ (uv), iIj (uv) > iI ′ (uv), j

+∞ −∞ +∞ [i−∞ Ij (uv), iIj (uv)]∩ [iI ′ (uv), iI ′ (uv)] = ∅ j

j

j

(uv) > fI−∞ • IVN fuzzy Ij −f bridge, if fI−∞ (uv) > fI+∞ (uv), fI+∞ (uv), ′ ′ j j j

[fI−∞ (uv), fI+∞ (uv)]∩ [fI−∞ (uv), fI+∞ (uv)] = ∅ ′ ′ j j j

j

j

ˇ iv = (I, I1 , I2 ) of graph structure G = Example 2.27. Consider an IVNGS G ˇ (U, R1 , R2 ) as shown in Fig. 12. Hiv = (I ′ , I1′ , I2′ ) is an IVN spanning subgraphˇ iv obtained by deleting an I1 -edge r2 r5 and shown in Fig. structure of IVNGS G 13. The edge r2 r5 is an IVN fuzzy I1 − bridge since −∞ +∞ +∞ • t−∞ I ′ (r2 r5 ) = 0.2, tI1 (r2 r5 ) = 0.7, tI ′ (r2 r5 ) = 0.3, tI1 (r2 r5 ) = 1

1

t+∞ I1 (r2 r5 ) = 0.8 > 0.3 = −∞ [tI ′ (r2 r5 ), t+∞ I1′ (r2 r5 )]= [0.7, 1

−∞ 0.8. t−∞ I1 (r2 r5 ) = 0.7 > 0.2 = tI ′ (r2 r5 ),

t+∞ I1′ (r2 r5 ),

[t−∞ I1 (r2 r5 ),

1

t+∞ I1 (r2 r5 )]

∩

0.8] ∩ [0.2, 0.3] = ∅. −∞ +∞ +∞ • i−∞ I ′ (r2 r5 ) = 0.3, iI1 (r2 r5 ) = 0.5, iI ′ (r2 r5 ) = 0.4, iI1 (r2 r5 ) = 1

1

−∞ 0.6. i−∞ I1 (r2 r5 ) = 0.5 > 0.3 = iI ′ (r2 r5 ),

i+∞ I1′ (r2 r5 ), +∞ [i−∞ I1 (r2 r5 ), iI1 (r2 r5 )] 0.4] = ∅.

1

i+∞ I1 (r2 r5 ) = 0.6 > 0.4 =

+∞ ∩ [i−∞ I ′ (r2 r5 ), iI ′ (r2 r5 )]= [0.5, 0.6] ∩ [0.3, 1

1

M. Akram and M. Sitara

I′ 1 ([ 0.

r1 ([0 .3, I′ 0.4 1 ([ 0.2 ], [ 0.6 , , [0 0.3 ′ ,0 .7, I2 ([0.2, 0.3], [0.4, 0.5], [0.3, 0.4]) ], .7] [0.

0.8 , [0 4, ], [ .4, 0 0.5 r3 ([0.5, 0.6], [0.7, 0.8], [0.5, 0.7]) .6 0.5 ], [ ,0 0.3 ]) .6] ) ,0 .4] )

126

r2 ([0.8, 0.9], [0.7, 0.8], [0.5, 0.7]) 2, 0.3 b ], [ I ′ ([0.3, 0.4], [0.6, 0.7], [0.4, 0.5]) 0.5 2 ,0 .6] , [0 .3, 0.4 ]) r5 ([0.7, 0.8], [0.5, 0.6], [0.6, 0.7]) b

b

b

I′ 1 ([ 0.

b

b

5, 0.6 ]

r6 ([0.3, 0.4], [0.4, 0.6], [0.4, 0.5]) ]) .7 ,0 .5 [0 ]) ], .4 .9 ,0 ,0 .3 .7 [0 [0 ], ], .4 .6 ,0 ,0 .5 .3 [0 ([0 r4

, 3] 0.

2, 0. ′ ([ I1

I2′ ([0.1, 0.2], [0.4, 0.5], [0.2, 0.3])

I2′ ([0.1, 0.2], [0.4, 0.5], [0.2, 0.3])

ˇ iv = (I ′ , I ′ , I ′ ) F IGURE 13. IVNGS H 1 2 • fI−∞ (r2 r5 ) = 0.3, fI−∞ (r2 r5 ) = 0.5, fI+∞ (r2 r5 ) = 0.4, fI+∞ (r2 r5 ) = ′ ′ 1 1 1

1

(r2 r5 ) = 0.5 > 0.3 = fI−∞ 0.7. fI−∞ (r2 r5 ) = 0.7 > 0.4 = (r2 r5 ), fI+∞ ′ 1 1 1

fI+∞ (r2 r5 ), [fI−∞ (r2 r5 ), fI+∞ (r2 r5 )] ∩ [fI−∞ (r2 r5 ), fI+∞ (r2 r5 )] = ′ ′ ′ 1 1 1 1 1 [0.5, 0.7] ∩ [0.3, 0.4] = ∅. ˇ iv = (I, I1 , I2 , . . . , It ) is an Ij -tree if (supp(I), supp(I1 ), Definition 2.28. An IVNGS G ˇ iv is an Ij -tree, if G ˇ iv supp(I2 ), . . . , supp(It )) is an Ij − tree. Alternatively, G has a subgraph induced by supp(Ij ) that forms a tree. ˇ iv = (I, I1 , I2 , . . . , It ) is an IVN fuzzy Ij -tree if Definition 2.29. An IVNGS G ˇ ˇ iv = (I ′′ , I ′′ , I ′′ , . . . , I ′′ ) Giv has an IVN fuzzy spanning subgraph-structure H 1 2 t ˇ iv , such that for all Ij -edges rs not in H ˇ iv is an I ′′ -tree and H j −∞ +∞ + + − (1) t− Ij (rs) < tI ′′ (rs), tIj (rs) < tI ′′ (rs), [tIj (rs), tIj (rs)] ∩ j

+∞ [t−∞ I ′′ (rs), tI ′′ (rs)] = ∅ j

j

j

−∞ +∞ + + − (2) i− Ij (rs) < iI ′′ (rs), iIj (rs) < iI ′′ (rs), [iIj (rs), iIj (rs)] ∩ j

j

+∞ [i−∞ I ′′ (rs), iI ′′ (rs)] = ∅ j

j

+ +∞ − + (3) fI−j (rs) < fI−∞ ′′ (rs), fI (rs) < fI ′′ (rs), [fI (rs), fI (rs)] ∩ j j j j

j

+∞ [fI−∞ ′′ (rs), fI ′′ (rs)] = ∅ j

j

In particular, + +∞ ˇ iv is an IVN fuzzy Ij −t tree if t− (rs) < t−∞ • G Ij I ′′ (rs), tIj (rs) < tI ′′ (rs), j

+ −∞ +∞ [t− Ij (rs), tIj (rs)] ∩ [tI ′′ (rs), tI ′′ (rs)] = ∅ j

j

j

[0 .4 ,0 .5 ],

[0 .5 ,0 .6 ])

I1

([0 I ([0.1, 0.2], [0.4, 0.5], [0.4, 0.5]) .3, 2 0.5 0.7]) r6 ], [ r5 ([0.4, 0.5], [0.5, 0.6], [0.6, ([ 0 0.4 .3, ,0 0.4 .5] , [0 ], [ 0.4 .4, ,0 0.5 .5] ]) ,[ [0.5, 0.6])

b

I2

b

r1 ([0.3, 0.4], [0.6, 0.7], [0.5, 0.6])

0.5 ]

)

[0.2, 0.3],

0.4 ,

I2 ([0.2, 0.3], [0.6, 0.7], [0.2, 0.3])

I1 ([0 .1, 0.2 ],

0.5 ] , [0 .4, 0.5 ] , [0 .4, 5, 0.6 ]

([0 .3 ,0 .4 ],

b

b

([ 0 I2 ([0.2, 0.3], [0.4, 0.5], [0.4, 0.5]) .3 ,0 .4 ], [ 0.

2, 0.

3], [0.

3, 0.

4])

I1

b

b

r4 ([0.5, 0.6], [0.7, 0.8], [0.5, 0.6])

r3 ([0.5, 0.6], [0.7, 0.8], [0.5, 0.6])

I1 ( [0.

127

I2 ([0.1, 0.2], [0.4, 0.5], [0.1, 0.2])

)

Interval-Valued Neutrosophic Graph Structures

r2 ([0.4, 0.5], [0.7, 0.8], [0.5, 0.6])

ˇ iv = (I, I1 , I2 ) F IGURE 14. G +∞ + ˇ iv is an IVN fuzzy Ij −i tree if i− (rs) < i−∞ • G Ij I ′′ (rs), iIj (rs) < iI ′′ (rs), j

j

+ −∞ +∞ [i− Ij (rs), iIj (rs)] ∩ [iIj′′ (rs), iIj′′ (rs)] = ∅ + ˇ iv is an IVN fuzzy Ij − f tree if f − (rs) < f −∞ • G Ij I ′′ (rs), fIj (rs) < j

− + −∞ +∞ fI+∞ ′′ (rs), [fI (rs), fI (rs)] ∩ [fI ′′ (uv), fI ′′ (uv)] = ∅ j j j

j

j

ˇ iv = (I, I1 , I2 ) of graph structure G = Example 2.30. Consider an IVNGS G (U, R1 , R2 ) as shown in Fig. 14. This IVNGS is I2 -tree, not I1 -tree. But it is ˇ iv IVN fuzzy I1 − t tree, since it has an IVN fuzzy spanning subgraph-structure H ′′ ′′ ′′ ′′ ˇ iv = (I , I1 , I2 ) as an I1 -tree, which is obtained by deleting I1 -edge r2 r5 from G and shown in Fig. 15. By direct calculations, we found that +∞ + − t−∞ I ′′ (r2 r5 ) = 0.3, tI ′′ (r2 r5 ) = 0.5, tI1 (r2 r5 ) = 0.1, tI1 (r2 r5 ) = 0.2, 1

1

−∞ + +∞ t− I1 (r2 r5 ) = 0.1 < 0.3 = tI ′′ (r2 r5 ), tI1 (r2 r5 ) = 0.2 < 0.5 = tI ′′ (r2 r5 ), 1

1

+∞ − + [t−∞ I ′′ (r2 r5 ), tI ′′ (r2 r5 )] ∩ [tI1 (r2 r5 ), tI1 (r2 r5 )] = [0.3, 0.5] ∩ [0.1, 0.2] = ∅.

1

1

ˇ iv1 = (I1 , I11 , I12 , . . . , I1t ) of graph structure G1 Definition 2.31. An IVNGS G ˇ iv2 = (I2 , I21 , I22 , . . . , I2t ) = (U1 , R11 , R12 , . . . , R1t ) is isomorphic to IVNGS G of graph structure G2 = (U2 , R21 , R22 , . . . , R2t ), if there is a pair (f, ϕ), where f : U1 → U2 is bijection and ϕ is a permutation on set {1, 2, . . . , t} such that: + − + − + − + [t− I1 (r), tI1 (r)] = [tI2 (f (r)), tI2 (f (r))], [iI1 (r), iI1 (r)] = [iI2 (f (r)), iI2 (f (r))], [fI−1 (r), fI+1 (r)] = [fI−2 (f (r)), fI+2 (f (r))], − + + [t− I1j (rs), tI1j (rs)] = [tI2ϕ(j) (f (r)f (s)), tI2ϕ(j) (f (r)f (s))], − + + [i− I1j (rs), iI1j (rs)] = [iI2ϕ(j) (f (r)f (s)), iI2ϕ(j) (f (r)f (s))],

[fI−1j (rs), fI+1j (rs)] = [fI−2ϕ(j) (f (r)f (s)), fI+2ϕ(j) (f (r)f (s))],

M. Akram and M. Sitara

0.5 ] , [0 .4, 0.5 ] 0.4 , 6], [ 5, 0 . b

([ 0 I2′′ ([0.2, 0.3], [0.4, 0.5], [0.4, 0.5]) .3 ,0 .4 ], [ 0.

2, 0.

3], [0.

3, 0.

4])

I

′ 1 ′(

[0 .3 ,0 .4 ],

[0 .4 ,0 .5 ],

[0 .5 ,0 .6 ])

I ′′ ′′ 1 ([ 0.3 I2 ([0.1, 0.2], [0.4, 0.5], [0.4, 0.5]) ,0 .5] r5 ([0.4, 0.5], [0.5, 0.6], [0.6, 0.7]) r6 ([ 0 , [0 .3, .4, 0.4 0.5 ], [ ], [ 0.4 0.4 ,0 ,0 .5] .5] , [0 ) . b

b

I2′′ ([0.2, 0.3], [0.6, 0.7], [0.2,

4, 0 0.3]) .5])

b

b

I1′′ ([0 .

I2′′ ([0.1, 0.2], [0.4, 0.5], [0.1, −0.2])

r3 ([0.5, 0.6], [0.7, 0.8], [0.5, 0.6])

r4 ([0.5, 0.6], [0.7, 0.8], [0.5, 0.6])

)

128

r2 ([0.4, 0.5], [0.7, 0.8], [0.5,

0.6])

b

I

2

′′

r1 ([0.3, 0.4], [0.6, 0.7], [0.5, 0.6])

ˇ iv = (I ′′ , I ′′ , I ′′ ) F IGURE 15. H 1 2 r4 ([0.3, 0.4], [0.2, 0.3], [0.5, 0.6]) , 0.

, [0 .2,

3 ],

0.3 ]

[0.2

, [0 .5,

, 0.

3 ],

0.6 ]

[0.5

, 0.

([0 .3 ,0 .4 ], [0 .3 ,0 r3 .4 ([0 ], [0 .3 .4 ,0 ,0 .4 .5 ], ]) [0 .7 ,0 .8 ], [0 .5 ,0 .6 ])

3, 0.4 ]

6])

)

[0.

I 1(

2, 0

, [0 .3 ]

.3 ,

0 .4

. ], [0

4, 0

b

I1

b

I1 ([0.3, 0.4], [0.2, 0.3], [0.4, 0.5])

I2 ( [0.2

([0 .

r2 ([0.6, 0.7], [0.5, 0.6], [0.5, 0.6])

b

I2

)

.5 ]

b

r1 ([0.4, 0.5], [0.3, 0.4], [0.4, 0.5])

ˇ iv1 = (I, I1 , I2 ) F IGURE 16. IVNGS G for all r ∈ U1 , rs ∈ I1j , j ∈ {1, 2, . . . , t}. ˇ iv1 = (I, I1 , I2 ) and G ˇ iv2 = (I ′ , I ′ , I ′ ) be two IVNGSs of Example 2.32. Let G 1 2 ′ two GSs G1 = (U, R1 , R2 ) and G2 = (U , R1′ , R2′ ) as shown in Fig. 16 and Fig. 17, respectively. ˇ iv1 and G ˇ iv2 are isomorphic under (f, ϕ), where f : U → U ′ is bijection and G ϕ is permutation on set {1, 2} defined as ϕ(1) = 2, ϕ(2) = 1, such that: + − + [t− I (ri ), tI (ri )] = [tI ′ (f (ri )), tI ′ (f (ri ))],

Interval-Valued Neutrosophic Graph Structures

129

s1 ([0.4, 0.5], [0.3, 0.4], [0.4, 0.5]) b

I ′ I2 ([0 2 ([ .3 , 0.

0 .4 2, ], [0 0.

3] .3 , , [0 0 .4 .3 ], [0 ,0 .4 , .4 0 .5 ], [ ]) 0.

4, 0. ]) 5] 6 ) 0.

. ′ [0 I 1(

s3 ([0.3, 0.4], [0.7, 0.8], [0.5, 0.6])

I2′ ([0.3, 0.4], [0.2, 0.3], [0.4, 0.5])

, . .6 2, 0 ([0 [0. , s2 ] 0 .4

[0 3 ],

b

, .5 [0 ], 7 0.

, .5 [0 ], 6 0. b

I1 ′ ([0 .2 ,0 .3 ], [0 .2 ,0 .3 ], [0 .5 ,0 .6 ])

′

.5 ,

0 .6

])

3,

b

s4 ([0.3, 0.4], [0.2, 0.3], [0.5, 0.6])

ˇ iv2 = (I ′ , I ′ , I ′ ) F IGURE 17. IVNGS G 1 2

+ − + [i− I (ri ), iI (ri )] = [iI ′ (f (ri )), iI ′ (f (ri ))],

[fI− (ri ), fI+ (ri )] = [fI−′ (f (ri )), fI+′ (f (ri ))], − + + [t− Ij (ri rk ), tIj (ri rk )] = [tIϕ(j) (f (ri )f (rk )), tIϕ(j) (f (ri )f (rk ))], + − + [i− Ij (ri rk ), iIj (ri rk )] = [iIϕ(j) (f (ri )f (rk )), iIϕ(j) (f (ri )f (rk ))],

[fI−j (ri rk ), fI+j (ri rk )] = [fI−ϕ(j) (f (ri )f (rk )), fI+ϕ(j) (f (ri )f (rk ))], for all ri ∈ U , ri rk ∈ Ij , j ∈ {1, 2} and i, k ∈ {1, 2, 3, 4}. ˇ iv1 = (I1 , I11 , I12 , . . . , I1t ) of graph structure G1 Definition 2.33. An IVNGS G ˇ iv2 = (I2 , I21 , I22 , . . . , I2t ) of = (U1 , R11 , R12 , . . . , R1t ) is identical to IVNGS G graph structure G2 = (U2 , R21 , R22 , . . . , R2t ) if f : U1 → U2 is a bijection, such that + − + − + − + [t− I1 (r), tI1 (r)] = [tI2 (f (r)), tI2 (f (r))], [iI1 (r), iI1 (r)] = [iI2 (f (r)), iI2 (f (r))], [fI−1 (r), fI+1 (r)] = [fI−2 (f (r)), fI+2 (f (r))], + − + [t− I1j (rs), tI1j (rs)] = [tI2j (f (r)f (s)), tI2j (f (r)f (s))], + − + [i− I1j (rs), iI1j (rs)] = [iI2j (f (r)f (s)), iI2j (f (r)f (s))],

[fI−1j (rs), fI+1j (rs)] = [fI−2j (f (r)f (s)), fI+2j (f (r)f (s))], for all r ∈ U1 , rs ∈ U1j , j ∈ {1, 2, . . . , t}. ˇ iv1 = (I, I1 , I2 ) and G ˇ iv2 = (I ′ , I1′ , I2′ ) be two IVNGSs of Example 2.34. Let G the graph structures G1 = (U, R1 , R2 ) and G2 = (U ′ , R1′ , R2′ ), respectively as shown in Fig. 18 and Fig. 19, respectively.

130

M. Akram and M. Sitara

], [ 0.5

, 0.2

.4, ([ ([0 I2 I1 I1 ([0.3, 0.4], [0.3, 0.4], [0.4, 0.5])

. ([0 I1

4, 0.5 ]

, [0 .4,

0.5 ]

, [0 .2,

0.3 ]

b

)

], [ 0.4

)

, 0.2

], [ 0.3

.5] ,0 0.4

)

I2 ( [0.

6, 0.7 ]

, [0 .4, 0.

I2 ([0.4, 0.5], [0.3, 0.4], [0.5, 0.6]) 5], [0 .5,

0.6 ]

) b

b

([0 .

3,

.6] ,0 0.5

r1 ([0.6, 0.7], [0.6, 0.7], [0.8, 0.9]) r7 ([0.6, 0.7], [0.6, 0.7], [0.6,

b

I1

], [ 0.3

], [ 0.6 5,

I

. ([0 2

.6, , [0

], [ 0.3 2,

.3] ,0 0.2

.3, , [0

]) 0.4

I

b

I

. ([0 2

.4] ,0 0.3

]) 0.7

r8 ([0.5, 0.6], [0.6, 0.7], [0.6, 0.7])

. ([0 2

] 0.7 6,

] 0.6 .5, , [0

,[

0.7]) ) .6] ,0 0.5

r5 ([0.6, 0.7], [0.6, 0.7], [0.5, 0.6])

)

b

.3, , [0

] 0.6 5,

b

.3] ,0 0.2

0. ], [ 0.4

b

r2 ([0.4, 0.5], [0.4, 0.5], [0.5, 0.6])

I2 ([0.3, 0.4], [0.3, 0.4], [0.4, 0.5]) r6 ([0.5, 0.6], [0.5, 0.6], [0.5, 0.6]) r3 ([0.6, 0.7], [0.4, 0.5], [0.6, 0.7])

r4 ([0.7, 0.8], [0.5, 0.6], [0.7, 0.8])

ˇ iv1 F IGURE 18. IVNGS G

] 0.3 2,

6, 0.7 ]

, [0 .4,

0.5 ]

, [0 .5,

0.6 ]

.2, , [0

I ′( 2 [0 .

0.7 ]

)

I2′ ([0.4, 0.5], [0.3, 0.4], [0.5, 0.6])

s1 ([0.4, 0.5], [0.4, 0.5], [0.5, 0.6]) s2 ([0.6, 0.7], [0.6, 0.7], [0.8, b

b

I′ 2 ([0 .

]) 0.3

5, 0.6 ], [ ] 0.3 5 .

0 ,0 , 4 .4] .

0 [ , [0 , ′ ([0.3, ] 5 .6, I 0.4], [0.3, 0.4], [0.4, 0.5]) 1 0.

], [ 0.3

.4, ′ ([0 I1

I2 I1′ ([0.3, 0.4], [0.3, 0.4], [0.4, 0.5])

)

) b

′

. ([0

] 0.4 3,

.2, , [0 ′

. ([0

I2

s4 ([0.6, 0.7], [0.4, 0.5], [0.6, 0.7])

. ,0 0.4

] 0.3 2,

,[

.4] ,0 0.3

.5, , [0

]) 0.6

′

b

I1

], [ 0.3

5])

. ([0

] 0.5 4,

.2, , [0

] 0.3

,[

0.9]) ) .6] ,0 0.5

b

s7 ([0.6, 0.7], [0.6, 0.7], [0.6, 0.7])

. ′ ([0 I2

.2, , [0

s6 ([0.5, 0.6], [0.6, 0.7], [0.6, 0.7])

.4] ,0 0.3

b

,[

)

b

′

] 0.7 .6, ([0

,[ .6] ,0 0.5

.6] ,0 0.5

b

s5 ([0.6, 0.7], [0.6, 0.7], [0.5, 0.6])

s3 ([0.7, 0.8], [0.5, 0.6], [0.7, 0.8])

I1

s8 ([0.5, 0.6], [0.5, 0.6], [0.5, 0.6])

ˇ iv2 F IGURE 19. IVNGS G

ˇ iv1 is identical to G ˇ iv2 under f : U → U ′ defined as : IVNGS G f (r1 ) = s2 , f (r2 ) = s1 , f (r3 ) = s4 , f (r4 ) = s3 , f (r5 ) = s5 , f (r6 ) = s8 , f (r7 ) = s7 , f (r8 ) = s6 . Moreover, + − + [t− I (ri ), tI (ri )] = [tI ′ (f (ri )), tI ′ (f (ri ))], + − + [i− I (ri ), iI (ri )] = [iI ′ (f (ri )), iI ′ (f (ri ))],

[fI− (ri ), fI+ (ri )] = [fI−′ (f (ri )), fI+′ (f (ri ))], − + + [t− Ij (ri rk ), tIj (ri rk )] = [tI ′ (f (ri )f (rk )), tI ′ (f (ri )f (rk ))], j

j

− + + [i− Ij (ri rk ), iIj (ri rk )] = [iI ′ (f (ri )f (rk )), iI ′ (f (ri )f (rk ))], j

j

[fI−j (ri rk ), fI+j (ri rk )] = [fI−′ (f (ri )f (rk )), fI+′ (f (ri )f (rk ))], j

j

for all ri ∈ U , ri rk ∈ Rj , j ∈ {1, 2}, i, k ∈ {1, 2, . . . , 8}.

Interval-Valued Neutrosophic Graph Structures

131

ˇ iv = (I, I1 , I2 , . . . , It ) be an IVNGS and ϕ be a permutaDefinition 2.35. Let G tion on {I1 , I2 , . . . , It } and also on the set {1, 2, . . . , t}, that is, ϕ(Ij ) = Il iff ϕ(j) = l for all j. If rs ∈ Ij and W − − (rs)] = [t− (rs), t+ [t− tϕ(Il ) (rs), I (r) ∧ tI (s) − Ijϕ Ijϕ l6=j W + − + t+ t+ I (r) ∧ tI (s) − ϕ(Il ) (rs)], [iIjϕ (rs), iIjϕ (rs)] = l6=j W + W − + − iϕ(Il ) (rs)], iϕ(Il ) (rs), i+ [i− I (r) ∧ iI (s) − I (r) ∧ iI (s) − l6=j

l6=j

[fI− (r) ∧ fI− (s) −

W

l6=j

[fI−ϕ (rs), fI+ϕ (rs)] = j

j

− fϕ(I (rs), fI+ (r) ∧ fI+ (s) − l)

W

l6=j

+ fϕ(I (rs)], l)

j = 1, 2, . . . , t, then rs ∈ Iuϕ , where u is selected, such that − + − + + − • t− ϕ (rs) ≥ t ϕ (rs), t ϕ (rs) ≥ t ϕ (rs), [t ϕ (rs), t ϕ (rs)] ∩ [ t ϕ (rs), I Iu Iu I Iu I Iu

(rs)] = ∅ t+ Iϕ

j

j

j

j

− + + − + − • i− ϕ (rs) ≥ i ϕ (rs), i ϕ (rs) ≥ i ϕ (rs), [i ϕ (rs), i ϕ (rs)] ∩ [ i ϕ (rs), Iu I Iu I Iu Iu I

(rs)] = ∅ i+ Iϕ

j

j

j

j

• fI−uϕ (rs) ≥ fI−ϕ (rs), fI+uϕ (rs) ≥ fI+ϕ (rs), [fI−uϕ (rs), fI+uϕ (rs)] ∩ [ fI−ϕ (rs), j

j

j

fI+ϕ (rs)] = ∅ j

for all j. Then IVNGS (I, I1ϕ , I2ϕ , . . . , Itϕ ) is said to be ϕ-complement of IVNGS ˇ iv and denoted by G ˇ ϕc .

G iv

Example 2.36. Let I = {(r1 , [0.4, 0.5], [0.4, 0.5], [0.7, 0.8]), (r2, [0.6, 0.7], [0.6, 0.7], [0.4, 0.5]), (r3 , [0.8, 0.9], [0.5, 0.6], [0.3, 0.4])}, I1 = {(r1 r3 , [0.4, 0.5], [0.4, 0.5], [0.3, 0.4])}, I2 = {(r2 r3 , [0.6, 0.7], [0.4, 0.5], [0.3, 0.4])}, I3 = {(r1 r2 , [0.4, 0.5], [0.3, 0.4], [0.4, 0.5])} be IVN subsets of U = {r1 , r2 , r3 }, R1 = {r1 r3 }, R2 = {r2 r3 }, R3 = ˇ iv = (I, I1 , I2 , I3 ) is an IVNGS of GS G = {r1 r2 }, respectively. Obviously, G (U, R1 , R2 , R3 ) as shown in Fig. 20

r2 ([0.6, 0.7], [0.6, 0.7], [0.4, 0.5]) r3 ([0.8, 0.9], [0.5, 0.6], [0.3, 0.4]) I2 ([0.6, 0.7], [0.4, 0.5], [0.3, 0.4]) b

] 0.4

5], [0.

4,

.3, , [0

0.5 ]

] 0.5 4,

, [0 .3,

. ([0 I3

0.4 ]

)

b

] 0.5 )

I1 ( [0.

4, 0 .

.4, , [0 b

r1 ([0.4, 0.5], [0.4, 0.5], [0.7, 0.8])

ˇ iv = (I, I1 , I2 , I3 ) F IGURE 20. G

132

M. Akram and M. Sitara

Simple calculations of edges r1 r3 , r2 r3 , r1 r2 ∈ I1 , I2 , I3 , respectively, show that ˇ ϕc =(I, I ϕ , I ϕ , I ϕ ) is ϕ-complement of r1 r3 ∈ I3ϕ , r2 r3 ∈ I1ϕ , r1 r2 ∈ I2ϕ . So, G 1 2 3 iv ˇ iv as shown in Fig. 21.

IVNGS G

] 0.4

I2ϕ( [0.4 , 0.

.3, , [0 ) b

I3ϕ ([0.4, 0.5], [0.4, 0.5], [0.3, 0.4])

r3 ([0.8, 0.9], [0.5, 0.6], [0.3, 0.4])

] 0.6

5], [0.4 ,

.5, , [0

0.5 ],

] 0.7 6,

[0.4 , 0.

. ([0

b

r1 ([0.4, 0.5], [0.4, 0.5], [0.7, 0.8])

ϕ

I1

b

5])

r2 ([0.6, 0.7], [0.6, 0.7], [0.4, 0.5])

ˇ iv = (I, I ϕ , I ϕ , I ϕ ) F IGURE 21. G 1 2 3 ˇ iv = (I, I1 , I2 , . . . , It ) is a strong Proposition 2.37. ϕ-complement of an IVNGS G IVNGS. Moreover, if ϕ(j) = u, where j, u ∈ {1, 2, . . . , t}, then all Iu -edges in IVNGS (I, I1 , I2 , . . . , It ) become Ijϕ -edges in (I, I1ϕ , I2ϕ , . . . , Itϕ ).

Proof. By definition of ϕ-complement, − (rs)] = [t− (rs), t+ [t− I (r) ∧ tI (s) − Iϕ Iϕ j

j

− (rs)] = [i− (rs), i+ [i− I (r) ∧ iI (s) − Iϕ Iϕ j

j

_

+ + t− ϕ(Il ) (rs), tI (r) ∧ tI (s) −

_

l6=j

l6=j

_

+ + i− ϕ(Il ) (rs), iI (r) ∧ iI (s) −

_

l6=j

[fI−ϕ (rs), fI+ϕ (rs)] = [fI− (r) ∧ fI− (s) − j

j

_

t+ ϕ(Il ) (rs)],(2. 1) i+ ϕ(Il ) (rs)],(2. 2)

l6=j − fϕ(I (rs), fI+ (r) ∧ fI+ (s) − l)

_

+ fϕ(I (rs)],(2. 3) l)

l6=j

l6=j

for j ∈ {1, 2, . . . , t}. For expression of truth membership value: W W − − + + t+ t (rs) ≥ 0, As t− (r) ∧ t (s) ≥ 0, t (r) ∧ t (s) ≥ 0 and I I I I ϕ(Il ) (rs) ≥ ϕ(Il ) l6=j

l6=j

− + + + 0. Since t− (rs) ≤ t− I (r) ∧ tI (s), tIj (rs) ≤ tI (r) ∧ tI (s), for all Ij . This W + WIj − + − − tϕ(Il ) (rs) ≤ t+ tϕ(Il ) (rs) ≤ tI (r) ∧ tI (s) and implies I (r) ∧ tI (s). It l6=j l6=j W + W − − tϕ(Il ) (rs) ≥ 0. Hence tϕ(Il ) (rs) ≥ 0, t+ shows that t− I (r) − I (r) ∧ tI (r) − l6=j

l6=j

(rs) obtain (rs) and t+ (rs) ≥ 0, for all j. Furthermore, t− (rs) ≥ 0 and t+ t− Ijϕ Ijϕ Ijϕ Ijϕ W + W − tϕ(Il ) (rs) are zero. Obviously, when tϕ(Il ) (rs) and maximum value when l6=j l6=j W + W − tϕ(Il ) (rs) acquire ϕ(Ij ) = Iu and rs is an Iu -edge then tϕ(Il ) (rs) and l6=j

l6=j

zero value. Hence + − + (rs)] = [t− (rs), t+ [t− I (r) ∧ tI (s), tI (r) ∧ tI (s)], f or (rs) ∈ Iu , ϕ(Ij ) = Iu .(2. 4) Iϕ Iϕ j

j

Interval-Valued Neutrosophic Graph Structures

133

For expression of indeterminacy membership value: W − W + − + + As i− iϕ(Il ) (rs) ≥ 0, iϕ(Il ) (rs) ≥ I (r) ∧ iI (s) ≥ 0, iI (r) ∧ iI (s) ≥ 0 and l6=j

l6=j

− + + + 0. Since i− (rs) ≤ i− I (r) ∧ iI (s), iIj (rs) ≤ iI (r) ∧ iI (s), for all Ij . This WIj − W + − − + implies iϕ(Il ) (rs) ≤ iI (r) ∧ iI (s) and iϕ(Il ) (rs) ≤ i+ I (r) ∧ iI (s). It l6=j l6=j W + W − − iϕ(Il ) (rs) ≥ 0. Hence iϕ(Il ) (rs) ≥ 0, i+ shows that i− I (r) − I (r) ∧ iI (r) − l6=j

l6=j

(rs) achieve (rs) and i+ (rs) ≥ 0, for all j. Furthermore, i− (rs) ≥ 0 and i+ i− Ijϕ Ijϕ Ijϕ Ijϕ W + W − iϕ(Il ) (rs) are zero. Obviously, when iϕ(Il ) (rs) and maximum value when l6=j l6=j W + W iϕ(Il ) (rs) get zero i− ϕ(Ij ) = Iu and rs is an Iu -edge then ϕ(Il ) (rs) and l6=j

l6=j

value. Hence − + + (rs)] = [i− (rs), i+ [i− I (r) ∧ iI (s), iI (r) ∧ iI (s)], f or (rs) ∈ Iu , ϕ(Ij ) = Iu .(2. 5) Iϕ Iϕ j

j

For expression of falsity membership value: W W + − As fI− (r)∧fI− (s) ≥ 0, fI+ (r)∧fI+ (s) ≥ 0 and fϕ(I (rs) ≥ 0, fϕ(Il ) (rs) ≥ l) l6=j

l6=j

0. Since fI−j (rs) ≤ fI− (r) ∧ fI− (s), fI+j (rs) ≤ fI+ (r) ∧ fI+ (s), for all Ij . This W − W + implies fϕ(Il ) (rs) ≤ fI− (r) ∧ fI− (s) and fϕ(Il ) (rs) ≤ fI+ (r) ∧ fI+ (s). It l6=j l6=j W + W − (rs) ≥ 0. Hence (rs) ≥ 0, fI+ (r)− fϕ(I shows that fI− (r)∧fI− (r)− fϕ(I l) l) l6=j

l6=j

fI−ϕ (rs) ≥ 0 and fI+ϕ (rs) ≥ 0, for all j. Furthermore, fI−ϕ (rs) and fI+ϕ (rs) obtain j j j j W + W − fϕ(Il ) (rs) are zero. Obviously, when fϕ(Il ) (rs) and maximum value when l6=j l6=j W + W − fϕ(Il ) (rs) acquire fϕ(I (rs) and ϕ(Ij ) = Iu and rs is an Iu -edge then l) l6=j

l6=j

zero value. Hence [fI−ϕ (rs), fI+ϕ (rs)] = [fI− (r) ∧ fI− (s), fI+ (r) ∧ fI+ (s)], f or (rs) ∈ Iu , ϕ(Ij ) = Iu .(2. 6) j

j

From expressions (2.4), (2.5) and (2.6), it is clear that − − − − − t− j (rs) = min{t (r), t (d)}, ij (rs) = min{i (r), i (s)}, − + − − + fj (rs) = min{f (−), f (s)}, tj (rs) = min{t (r), t+ (s)}, + + + + + i+ j (rs) = min{i (r), i (s)}, fj (rs) = min{f (r), f (s)},

ˇ iv is a strong IV N GS and all Iu -edges in IVNGS (I, I1 , I2 , . . . , It ) Hence G

become Ijϕ -edges in (I, I1ϕ , I2ϕ , . . . , Itϕ ). ˇ iv = (I, I1 , I2 , . . . , It ) be an IVNGS and ϕ be a permutaDefinition 2.38. Let G tion on {1, 2, . . . , t}. Then ˇ iv is self complementary IVNGS if G ˇ iv is isomorphic to G ˇ ϕc . (i) G iv ˇ iv is strong-self complementary IVNGS if G ˇ iv is identical to G ˇ ϕc . (ii) G iv ˇ iv = (I, I1 , I2 , . . . , It ) be an IVNGS. Then Definition 2.39. Let G ˇ iv is totally-self complementary IVNGS if G ˇ iv is isomorphic to G ˇ ϕc , (i) G iv for all permutations ϕ on {1, 2, . . . , t}.

134

M. Akram and M. Sitara

ˇ iv is totally-strong-self complementary IVNGS if G ˇ iv is identical to (ii) G ϕc ˇ Giv , for all permutations ϕ on {1, 2, . . . , t}. ˇ iv = (I, I1 , I2 , I3 ) shown in Fig. 22 is identical to Example 2.40. An IVNGS G ϕ−complement for all permutations ϕ on set {1, 2, 3}. Hence it is totally-strongself complementary IVNGS. 4], [0 .4, 0.

I2 ([0 5], [0 .3, 0 .5, 0. .4], [ 6]) 0.4, 0 I3 .5], [ ([0 r7 ( 0.5, 0 I3 [0. .3 ( .6]) ,0 3, [0.

0.4 .4 3, ], [ ], [ 0.4 0.3 0. ] , 3, [ , 0.

0. r6 ([ 0.3, 4], 4] 0.5 0.4 [0. , [0 4, 0 , 0 ], [ . .5 6], 0.4 .5] ,0 ) [0. , 0. .6 5, 5]) ]) 0.6 ], [ 0.6 ,0 .7] )

) 0.6] 0.5, 6], [ 5, 0.

6]) , [0.

5, 0.

0.6] ], [0.

0.5, , 0.5 r 4([ ]) , [0.4 0.6] ) 0.7 0.5, 6, .6] ]) 0 I 2([ [0. , .5 .5, .6] ,0 , [0 ,0 .4 .5] [0 0.5 ,0 ], ], [ .4 0.4 ]) 0.6 5 . ,0 ], [ 0 .5, .3 4, ([0 0.6 [0 0. r2 ], .5, ,[ .4 ([0 4] . ,0 I1 .3 ,0 .3 ([0 [0 I1

b

b

r5 ([0 .3, 0.

b

], .4 ,0 .3 ([0 r3

b

b

b

b

r1 ([0.8, 0.9], [0.4, 0.5], [0.5, 0.6])

F IGURE 22. Totally-strong-self complementary IVNGS

Theorem 2.41. An IVNGS is totally-self complementary if and only if it is a strong IVNGS. ˇ iv and permutation ϕ on {1,2, . . . , t}. By Proof. Consider a strong IVNGS G ˇ iv = (I, I1 , I2 , . . . , It ) is a strong proposition 2.37, ϕ-complement of IVNGS G −1 IVNGS. Moreover, if ϕ (u) = j, where j, u ∈ {1, 2, . . . , t}, then all Iu -edges in IVNGS (I, I1 , I2 , . . . , It ) become Ijϕ -edges in (I, I1ϕ , I2ϕ , . . . , Itϕ ), this leads − − − − − − − t− Iu (rs)=tI (r) ∧ tI (s)=tI ϕ (rs), iIu (rs)=iI (r) ∧ iI (s)=iI ϕ (rs), j

j

+ + + fI−u (rs)=fI− (r) ∧ fI− (s)=fI−ϕ (rs), t+ Iu (rs)=tI (r) ∧ tI (s)=tI ϕ (rs), j

j

+ + + + + + + i+ Iu (rs)=iI (r) ∧ iI (s)=iI ϕ (rs), fIu (rs)=fI (r) ∧ fI (s)=fI ϕ (rs). j

j

ˇ iv and Therefore, under f : U → U (identity mapping), G such that:

ˇϕ G iv

are isomorphic

− − − − − t− I (r) = tI (f (r)), iI (r) = iI (f (r)), fI (r) = fI (f (r)), + + + + + + tI (r) = tI (f (r)), iI (r) = iI (f (r)), fI (r) = fI (f (r)). + + + − − t− Iu (rs) = tI ϕ (f (r)f (s)) = tI ϕ (rs), tIu (rs) = tI ϕ (f (r)f (s)) = tI ϕ (rs),

=

(f (r)f (s)) i− Ijϕ

=

(rs), i− Ijϕ

j

j

j

j

i− Iu (rs)

i+ Iu (rs)

=

(f (r)f (s)) i+ Ijϕ

=

(rs), i+ Ijϕ

fI−u (rs) = fI−ϕ (f (r)f (s)) = fI−ϕ (rs), fI+u (rs) = fI+ϕ (f (r)f (s)) = fI+ϕ (rs), j

j

j

j

Interval-Valued Neutrosophic Graph Structures

135

for all rs ∈ Iu , for ϕ−1 (u) = j; j, u = 1, 2, . . . , t. ˇ iv is totally-self This holds for every permutation ϕ on {1, 2, . . . , t}. Hence G ϕ ˇ ˇ complementary IVNGS. Conversely, let Giv is isomorphic to Giv for each permutation ϕ on {1, 2, . . . , t}. Moreover, according to the definitions of isomorphism of IVNGSs and ϕ-complement of an IVNGS − − − − − t− Iu (rs) = tI ϕ (f (r)f (s)) = tI (f (r)) ∧ tI (f (s)) = tI (r) ∧ tI (s), j

+ + + + + t+ Iu (rs) = tI ϕ (f (r)f (s)) = tI (f (r)) ∧ tI (f (s)) = tI (r) ∧ tI (s), j

i− Iu (rs) i+ Iu (rs)

= =

(f (r)f (s)) i− Ijϕ + iI ϕ (f (r)f (s)) j

− − − = i− I (f (r)) ∧ iI (f (s)) = iI (r) ∧ iI (s), + + + = i+ I (f (r)) ∧ iI (f (s)) = iI (r) ∧ iI (s),

fI−u (rs) = fI−ϕ (f (r)f (s)) = fI− (f (r)) ∧ fI− (f (s)) = fI− (r) ∧ fI− (s), j

fI+u (rs) = fI+ϕ (f (r)f (s)) = fI+ (f (r)) ∧ fI+ (f (s)) = fI+ (r) ∧ fI+ (s), j

ˇ iv is a strong IVNGS. for all rs ∈ Iu , u = 1, 2, . . . , t. Hence G

Remark. Every self complementary IVNGS is totally-self complementary.

Theorem 2.42. If G = (U, R1 , R2 , . . . , Rt ) is a totally-strong-self complementary graph structure and + − + + − − + − − + + I = ([t− I , tI ], [iI , iI ], [fI , fI ]) is an IVN subset of U where tI , iI , fI , tI , iI , fI are constant functions, then every strong IVNGS of G with IVN vertex set I is a totally-strong-self complementary IVNGS. Proof. Let a, a′ ∈ [0, 1], b, b′ ∈ [0, 1] and c, c′ ∈ [0, 1] be six constants and + + − − ′ ′ ′ + t− I (r) = a, iI (r) = b, fI (r) = c, tI (r) = a , iI (r) = b , fI (r) = c , for all r ∈ U .

Since G is a totally-strong-self complementary GS, so for every permutation ϕ−1 on {1, 2, . . . , t} there is a bijection f : U → U , such that for every Iu −edge (rs), −1 (f(r)f(s)) [an Ij -edge in G ] is an Iu -edge in Gs ϕ c . Thus for every Iu -edge (rs), −1 ˇ iv ] is an I ϕ -edge in Gˇiv ϕ c . (f(r)f(s)) [an Ij -edge in G u ˇ iv is a strong IVNGS, so Moreover, G − − − t− I (r) = a = tI (f (r)), iI (r) = b = iI (f (r)), + + + ′ ′ tI (r) = a = tI (f (r)), iI (r) = b = i+ I (f (r)),

fI− (r) = c = fI− (f (r)), fI+ (r) = c′ = fI+ (f (r)),

for all r ∈ U , and − − − − − t− Iu (rs) = tI (r) ∧ tI (s) = tI (f (r)) ∧ tI (f (s)) = tI ϕ (f (r)f (s)), j

− − − − − i− Iu (rs) = iI (r) ∧ iI (s) = iI (f (r)) ∧ iI (f (s)) = iI ϕ (f (r)f (s)), j

− − − fI−u (rs) = fI− (r) ∧ i− I (s) = fI (f (r)) ∧ fI (f (s)) = fI ϕ (f (r)f (s)), j

+ + + + + t+ Iu (rs) = tI (r) ∧ tI (s) = tI (f (r)) ∧ tI (f (s)) = tI ϕ (f (r)f (s)), j

+ + + + + i+ Iu (rs) = iI (r) ∧ iI (s) = iI (f (r)) ∧ iI (f (s)) = iI ϕ (f (r)f (s)), j

+ + + fI+u (rs) = fI+ (r) ∧ i+ I (s) = fI (f (r)) ∧ fI (f (s)) = fI ϕ (f (r)f (s)), j

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M. Akram and M. Sitara

for all rs ∈ Ij , j = 1, 2, . . . , t. ˇ iv is a strong-self complementary IVNGS. This satisfies for each This shows G ˇ iv is a totally-strong-self compermutation ϕ and ϕ−1 on set {1, 2, . . . , t}, thus G plementary IVNGS. This completes the proof.

Remark. Converse of theorem 2.42 may not true, for example a IVNGS depicted in Fig. 22 is totally-strong-self complementary IVNGS, it is also strong IVNGS − with a totally-strong-self complementary underlying graph structure but t− I , iI , − + + + fI ,tI , iI , fI are not the constant functions.

3. C ONCLUSIONS Interval-valued fuzzy set theory has numerous applications in various fields of science and technology, including, fuzzy control, artificial intelligence, operations research and decision-making. An interval-valued neutrosophic graph constitutes a generalization of the notion interval-valued fuzzy graph. In this research paper, we have introduced the notion of interval-valued neutrosophic graph structures and discussed many relevant notions with appropriate examples. We have also discussed some interesting properties of these notions.

Conflict of interest. The authors declare that they have no conflict of interest.

Acknowledgment. The authors are highly thankful to the referees for their valuable comments and suggestions.

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